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Bell's Theorem and Experimental Tests BACHELORARBEIT Titel der Bachelorarbeit Bell's theorem and experimental tests Verfasser Cheng Xiaxi Matrikelnummer: a0800812 Institut für Experimentalphysik der Universität Wien angestrebter akademischer Grad Bachelor of Science (B. Sc.) Wien, im Juli 2011 Studienkennzahl lt. Studienblatt: 033676 Betreuer: Ao. Univ. -Prof. Dr. Reinhold A. Bertlmann Contents 1 Introduction 3 2 Basic concepts of quantum mechanics 5 2.1 The density operator . .7 2.2 The entangled state . .8 3 Part I: Bell's Theorem 9 3.1 Bell's Theorem . .9 3.2 The general argument (1964) . 11 3.3 CHSH { The non-idealized case (1971) . 13 3.4 CH { more experiment related (1974) . 15 3.5 Wigner { a combinatorial analysis . 17 3.6 Relations between Bell's inequality and entanglement witness 18 4 Part II: Experimental Tests 20 4.1 Requirement for general experimental tests . 20 4.2 The \First" Generation . 22 4.3 The \Second" Generation . 24 4.4 The \Third" Generation - G. Weihs, A. Zeilinger 1998 . 27 4.5 Lab report . 29 4.5.1 Experimental requirements . 29 4.5.2 Setup and execution . 29 4.5.3 Results . 33 4.5.4 Discussion . 36 5 Conclusion 37 6 Acknowledgement 38 2 Chapter 1 Introduction We can say that one of the greatest philosophical and physical discussions was caused by a single paper written by Einstein, Podolsky and Rosen[1] (EPR) in 19351. In the paper, EPR assume reasonably the non-existence of the action-at-distance and define the criteria for a complete physical theory, which include the concepts of local causality and physical reality. By demon- strating mathematically their ideas within a though experiment, which was modified by Bohm[26] later on, they concluded that the description of quan- tum mechanics may be incomplete. This conclusion is formulated because this real, deterministic property isn't contained in the formalism of quan- tum theory. And that is also the birth of the term of the \spooky action at the distance[1]." The observation on one system immediately influences the observation on the second one. Few months later, Bohr[2] replied to EPR and defended quantum mechanics with his principle of complementarity. He denoted that the realistic view- point is inapplicable under the macroscopic level. In the following decades, in order to fulfill the desire of understanding na- ture, not only was the validation of contextuality in nature questioned, but also the suggestions of extending quantum physics with a model of unknown \hidden" variable theory (HVT) for the completion were proposed. Many great physicists, e.g. Von Neumann, Gleason, Jauch and Piron, made some attempts to deny the theoretical extension. Apparently, all the proofs con- tain some leakages in their axioms or failed at the point of determinism and congruence with quantum mechanics[3]. Finally, both possible explanations were fully eliminated by the two follow- ing \no-go" theorems: the Kochen-Specker theorem and Bell's theorem. The present work covers Bell's theorem, published in 1964[4] by John Bell. He formulates a contradiction between the HVT and quantum mechanical 1Interestingly, the EPR paper became cited with a delay of 30 years, i.e., after Bell published his inequality and nowadays in the age of quantum information, there is an exponential increase of citations. 3 measurement. Besides the brief summaries of a few interesting proofs of Bell's theorem, the most significant historical experimental tests of Bell's theorem, during the last decades, are hereby also described and analysed. 4 Chapter 2 Basic concepts of quantum mechanics In this chapter, a short introduction into quantum mechanics will be ex- plained and all necessary mathematical notations for understanding the present work will also be mentioned. According to the fundamental law of quantum mechanics, a quantum state φ of a given system is described by a complex function, called the wave func- tion, which contains the complete knowledge of the state and allows one to determine the probabilities of obtaining certain measurement results made in the state. Suppose φ(~x;t) is a function of the space ~x and time t, then the probability of finding the system in the coordinate interval [~x;~x + d~x] or shortly d3x at time t is given by jφ(~x;t)j2d3x (2.1) And due to the normalization condition, the total probability is: Z jjφjj2 = jφ(~x;t)j2d3x = 1 = hφjφi (2.2) where jj · jj = ph · j · i and the right side of the equation is expressed by the Dirac notation. To gain a deeper understanding of the quantum states, let us consider the space underlying our physical objects, which is also called the Hilbert space. Definition: A Hilbert space H is a finite- or infinite-dimensional, linear vector space with the scalar product in C (Complex field). The inner product of the elements of the space, hφj i = R φ∗ d3x, with φ, 2 H, satisfies the following properties, where φ∗ is the complex conjugate of φ: hφj i is complexly conjugated: hφj i = h jφi where the term with overline denotes the complex conjugation. 5 hφj i is linear in its second argument. For all complex number a and b: hφja 1 + b 2i = ahφj 1i + bhφj 2i The inner product is positive definite: hφjφi ≥ 0 where the case of equality holds for φ = 0. The norm, a real-value function, is also defined by the inner product: jjφjj = phφjφi Physical states expressed by wave functions obey the superposition princi- ples. If fjφiig are all possible states of H, then any linear combination jφi of them also belongs to H: jφi = c1jφ1i + c2jφ2i + ::: + cN jφN i (2.3) 2 where fcig are complex numbers and i = 1; 2;:::N. jcij gives the proba- PN 2 bility of obtaining φi with i=1 jcij = 1: 2 2 jcij = jhφij ij (2.4) If fjφiig are orthonormal, they are called the orthonormal bases. Therefore a quantum state can be generally written as: N X jφi = cijφii (2.5) i=1 The observables are operators which act on the wave function, if certain physical quantities need to be measured. The measurement results are given by the eigenstates of the corresponding observable with their eigenvalues. Imagine an operator A acting on any arbitrary eigenstate jφii, then: Ajφii = λijφii where λi is the eigenvalue and denotes the measurement probability. The most important analysis of this work is based on the expectation value of an observable. Since a statistical average value for the given observable acted on a large number of the identically prepared systems is required, this mean value is defined by the expectation value: Z N ∗ 3 X y hAiφ = φ Aφd x = hφjAjφi = cncpAn;p (2.6) n;p=0 where An;p = hφnjAjφpi is the matrix element of the observable A. 6 2.1 The density operator Usually, we often deal with states of systems, which are not perfectly de- termined. In order to make the maximal use of this partial information we possess about the state of the system, the density matrix formalism is introduced. The density operator ρ is therefore defined as: ρ = jφihφj (2.7) Analogously, the expectation value is given by: hAiφ = tr(ρ · A) = tr(A · ρ) (2.8) And tr(ρ) = 1 (2.9) For a given pure state jφi whose information is perfectly known, the density operator has to fulfil the following properties: ρ2 = ρ ρy = ρ tr(ρ2) = 1 For the completeness, a mixed state is present when there is a statistical distribution of pure states. Since there is no interference effects between the pure states, a mixed state can therefore only be given by the density operator: X X ρ = pijφiihφij = piρi (2.10) i i P where i pi = 1 and fpig are the probabilities of detecting fjφiihφijg. The following properties will be satisfied: X tr(ρ) = pitr(ρi) i tr(ρ) = 1 hAiφ = tr(ρ · A) ρ2 6= ρ tr(ρ2) ≤ 1 Here the last inequality can be seen as a measure of the mixedness of a given state. 7 2.2 The entangled state Before starting with Bell's theorem, let us first take a brief look at the entangled state. We consider the following quantum state vector: · · −iφ · jΦi = a jxi1 jyi2 + b e jyi1 jxi2 (2.11) which is a general form of an entangled state for two subsystems. Here, jaj2 2 and jbj are the probabilities of the corresponding product states jxi1⊗jyi2 = 2 2 −iφ jxi1 jyi2 and jyi1 ⊗jxi2 = jyi1 jxi2 with jaj +jbj = 1, the term e denotes a phase difference. The peculiar property of this kind of states is that they can't be factorized into a product of two states of individual subsystems. In other words, a state vector jΨi in biparticle system describing, e.g., two subsystems of photons can generally be written in the form: X jΨi = cm;n jmi1 jni2 (2.12) m;n · where cm;n = am bn are the probability coefficients of jmi1 jni2, am is the P m-th probability coefficient of the single photon state jΨ1i = am jmi, the m analogous notation bn for the other single photon. A state which can be written in Eq. (2.12) is called separable. Otherwise, the state is said to be entangled, which means: cm;n 6= am · bn (2.13) In conclusion, an entangled state can only be totally examined as a whole system. Therefore we define an entangled state as such a state with two or more subsystems, separated by a certain distance, that the whole system, but not the individual subsystems, is well defined. The most remarkable entangled states are called Bell states, which have maximal entanglement and are given for photons with horizontal and vertical polarizations by: + 1 Ψ = p (jHi jV i + jV i jHi ) (2.14) 2 1 2 1 2 − 1 Ψ = p (jHi jV i − jV i jHi ) (2.15) 2 1 2 1 2 + 1 Φ = p (jHi jHi + jV i jV i ) (2.16) 2 1 2 1 2 − 1 Φ = p (jHi jHi − jV i jV i ) (2.17) 2 1 2 1 2 8 Chapter 3 Part I: Bell's Theorem 3.1 Bell's Theorem Let us consider the following thought experiment made by Bohm[26] (EPRB).
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